Planet Occurrence Rates from Doppler and Transit Surveys

Transcription

1 Planet Occurrence Rates from Doppler and Transit Surveys Joshua N. Winn Abstract Prior to the 990s, speculations about the occurrence of planets around other stars were informed only by planet formation theory, observations of circumstellar disks, and the knowledge that at least one seemingly ordinary star the Sun had managed to make a variety of different planets. Since then, Doppler and transit surveys have exposed the population of planets around other Sun-like stars, especially those with orbital periods shorter than a few years. Over the last decade these surveys rose to a new level of perfection with Doppler spectrographs capable of m s precision, and space telescopes capable of detecting the transits of Earthsized planets. This article is a brief introductory review of the knowledge of planet occurrence that has been gained from these surveys. Introduction If, in some cataclysm, all our knowledge of exoplanets were to be destroyed, and only one sentence passed on to the next generation of astronomers, what statement would contain the most helpful information? This challenge, a modified version of the one posed by Richard Feynman in his Lectures on Physics, is difficult to meet with words. Here is one possibility: Most Sun-like stars have planets, which display a wider range of properties size, mass, orbital parameters than the planets of the Solar System. The job would be easier if we could convey a mathematical function. The occurrence rate density (or simply occurrence) is the expected number of planets per star that have certain properties. For example, transit observations reveal a planet s radius R and orbital period P. We can summarize the findings of a transit survey with a function Princeton University, Department of Astrophysical Sciences, 4 Ivy Lane, Princeton, NJ 08544, USA, jnwinn@princeton.edu

2 2 Joshua N. Winn Γ R,P d 2 N d logr d logp, () giving the mean number of planets per logarithmic interval of radius and period. A simple and accurate occurrence function would help our descendants design new instruments to detect planets, and inspire theories for planet formation. Occurrence depends on other planetary parameters, such as orbital eccentricity; and on the characteristics of the star, such as mass, metallicity, and age. Occurrence is also likely to depend on the properties of any other planets that exist in a particular system. No analytic function could possibly represent all these parameters and their correlations. Ideally we would transmit a computer program that produces random realizations of planetary systems that are statistically consistent with everything we have learned from planet surveys. What follows is an brief introductory review of the progress toward this goal that has come from Doppler and transit surveys. The cited works were chosen to be useful entry points to the literature, not to provide a comprehensive bibliography. The basics of the Doppler and transit methods themselves are left for other reviews, such as those by Lovis and Fischer (200), Winn (200), Wright (this volume) and Bakos (this volume). The next section describes methods for occurrence calculations. Surveys have shown major differences in occurrence between giant planets and small planets, with a dividing line of about 6 R or 30 M. Thus the results for giants and small planets are presented separately, in two sections. After that comes a review of what is known about other types of stars, followed by a discussion of future prospects. Methods Life would be simple if planets came in only one type, and we could detect them unerringly. We would search N stars, detect N det planets, and conclude that the integrated occurrence is Γ N det /N. But detection is not assured, because small signals can be lost in the noise. If the detection probability were p det in all cases, then we would have effectively searched only p det N stars, and the estimated occurrence would be N det /(p det N). In reality, p det depends strongly on the characteristics of the star and planet, as illustrated in Figure. Detection is easier for brighter stars, larger planets (relative to the star), and shorter orbital periods. For this reason we need to group the detected planets according to orbital period and other salient characteristics for detection: radius R, for transit surveys; and m M sini for Doppler surveys, where M is the mass and I is the orbital inclination. Then our estimate becomes N Γ i N d p det,i j, (2) j=

3 Planet Occurrence Rates 3 mass [M ] Mmin~P3/2 0 Mmin~P/3 Survey duration orbital period [days] 0000 Fig. Idealized Doppler survey of 04 identical Sun-like stars. Each star has one planet on a randomly-oriented circular orbit, with a mass and period drawn from log-uniform distributions between the plotted limits. Each star is observed 50 times over one year with m s precision. The small dots are all the planets; the large blue dots are those detected with 5σ confidence. For P shorter than the survey duration, the threshold mass is proportional to P/3, corresponding to a constant Doppler semi-amplitude K m s. For longer P, the threshold mass increases more rapidly, with an exponent depending on the desired confidence level (Cumming 2004). where the indices i and j specify the type of planet and star, respectively. Transit surveys have the additional problem that there is no signal at all unless cos I < R? /ac, where ac is the orbital distance at the time of inferior conjunction. Thus we must also divide by the probability for this condition to be met, which is equal to R? /a for randomly-oriented circular orbits. This conceptually simple method has been the basis of many investigations. The results of Doppler surveys are often presented as a matrix of Γ values for rectangular regions in the space of m M sin I and P; for transit surveys the regions are in the space of R and P. Ideally, each region should be large enough to contain many detected planets, and yet small enough that the detection probability does not vary too much from one side to the other. In practice these conditions are rarely achieved, and more complex methods are preferable. One approach is to posit a functional form for the occurrence rate density, such as a power law d2n Γm,P = = Cmα Pβ, (3) d ln m d ln P

4 4 Joshua N. Winn radius [R ] x survey duration Rmin ~ P/ orbital period [days] 0000 Fig. 2 Idealized transit survey of 04 identical Sun-like stars. Each star has one planet on a randomly-oriented circular orbit, with a radius and period drawn from log-uniform distributions between the plotted limits. Each star is observed continuously for one year with a photon-limited photometric precision corresponding to 00 parts per million over 6 hours. The small dots are all the planets; the large blue dots are those detected with 0σ confidence based on at least two transit detections. Compared to the Doppler survey, the transit survey finds fewer planets and is more strongly biased toward short periods. This is because the geometric transit probability is low and is proportional to P 2/3. The threshold radius varies as P/6 out to twice the survey duration (Pepper et al. 2003). and use this function to construct a likelihood function for the outcome of a survey. The likelihood function must take into account the detection probability, the properties of the detected systems, and the properties of the stars for which no planets were detected. Then the values of the adjustable parameters C, α, and β are determined by maximizing the likelihood. Details of this method are provided by Tabachnik and Tremaine (2002) or Cumming et al. (2008) for Doppler surveys, and Youdin (20) for transit surveys. Foreman-Mackey et al. (204) generalized this technique to cope with uncertainties in the planet properties such as R and P. They also cast the problem in the form of Bayesian hierarchical inference. Most studies report the probability for a star to have a planet with certain properties, regardless of any other planets in the system. Accounting for multiple-planet systems is more difficult. For Doppler surveys, the main problem is that the star is pulled by all the planets simultaneously. As a result, the detectability of a given signal to depend on the properties of any other detectable planets especially their periods and on the timespan and spacings between the data points. This makes it difficult to calculate the detection probability.

5 Planet Occurrence Rates 5 For transit surveys, the different planetary signals do not overlap very much. Instead, the problem is the degeneracy between multiplicity and inclination dispersion (Tremaine and Dong 202). A star with only one detected planet could lack additional planets, or it could have a system of many planets only one of which has an inclination close enough to 90 to exhibit transits. In principle this degeneracy can be broken by combining the results of Doppler and transit surveys and solving for both the occurrence rate density as well as the planetary mass-radius relationship. Doppler surveys have uncovered a total of about 500 planets. The most informative surveys for planet occurrence were based on observations with the High Resolution Echelle Spectrometer (HIRES) on the Keck I 0-meter telescope (Cumming et al. 2008; Howard et al. 200) and the High Accuracy Radial-velocity Planet Searcher (HARPS) on the La Silla 3.6-meter telescope (Mayor et al. 20). Both instruments were used to monitor 0 3 stars for about a decade, with a precision of a few meters per second. Additional information about wide-orbiting giant planets comes from a few lower-precision and longer-duration surveys. For transits, the ground-based surveys have discovered about 200 planets, but are not well suited to occurrence calculations because both the sample of searched stars and the detection probabilities are poorly characterized. Instead our most important source is the NASA Kepler mission, which used a -meter space telescope to measure the brightness of 50,000 stars every 30 minutes for 4 years (Borucki 206). The typical photometric precision over a 6-hour time interval was of order 00 parts per million (ppm). This was sufficient to detect several thousand planets. Giant planets Overall occurrence For giant planets the key references are Cumming et al. (2008) and Mayor et al. (20) for Doppler surveys, and Santerne et al. (206) for Kepler. These studies agree that giant planets with periods shorter than a few years are found around 0% of Sun-like stars (see Fig. 3). In particular Cumming et al. (2008) studied planets with a minimum mass m in the range from M Jup and P from days. They found the best-fitting power-law of the form given by Eqn. (3) to have α = 0.3 ± 0.20 and β = 0.26 ± 0.0, and a normalization such that 0.5% of Sun-like stars have such a planet. They also found the data to be equally well described by a distribution uniform in logp from days (i.e., β = 0), followed by a sharp increase by a factor of 4-5 for longer periods. The Kepler data are also consistent with this description (Santerne et al. 206). Hot Jupiters Hot Jupiters are the easiest type of planet to detect. However, they are also intrinsically rare, with an occurrence of 0.5-% for periods -0 days. For giant planets with even shorter periods, dn/d lnp is lower by at least an order of magnitude (Howard et al. 202; Sanchis-Ojeda et al. 204). There is a growing literature addressing the apparent 2-3σ discrepancy between the Hot Jupiter occurrence of % measured in Doppler surveys (Wright et al. 202; Mayor et al. 20) and

6 6 Joshua N. Winn the lower figure of 0.6% measured using Kepler data (Howard et al. 202; Petigura et al. 207b). Two aspects of the Kepler sample of stars that might be contributing to the difference are its slightly lower mean metallicity (Guo et al. 207) and higher binary fraction (Wang et al. 205) than the Doppler sample. Jupiter analogs A perennial question is whether the Solar System is typical or unusual in some sense. Several groups have used Doppler surveys to try and answer this question by measuring the occurrence of Jupiter-like planets. The idea is that Jupiter would be the most easily detected planet around the Sun in an extraterrestrial Doppler survey. The answer depends on the definition of Jupiter analog, a term without a precise meaning. (The same problem arises when trying to measure the occurrence of Earth-like planets.) Wittenmyer et al. (206) presented the latest effort, finding the occurrence to be % for planets of mass M Jup with orbital distances from 3-7 AU and eccentricities smaller than 0.3. Long-period giants Regarding the more general topic of wide-orbiting giant planets, Foreman-Mackey et al. (206) measured the occurrence of cold Jupiters through an automated search of the Kepler data for stars showing only one transit over 4 years. They concluded Γ R,P = d 2 N = ± (4) d lnr d lnp over the range of periods from 2-25 years and planetary radius R = 0.- R Jup. Early work by Wright et al. (2009) suggested that when one giant planet is detected in a Doppler survey, the odds of detecting a wider-orbiting companion within a few years are at least 30%. More recently Bryan et al. (206) used high-resolution imaging and long-term Doppler monitoring to search for wide-orbiting companions to 23 giant planets with orbital distances ranging from 0.0 to 5 AU. They found that for the outer companions, dn/d lnp is a declining function of period, unlike the more uniform distribution observed for the inner planets. They measured an occurrence of (53 ± 5)% for outer companions of mass -20 M Jup and orbital distances of 5-20 AU. This is about twice has high as the occurrence that was measured by Foreman-Mackey et al. (206) without regard to multiplicity, suggesting a linkage between the formation of inner and outer giant planets. Metallicity The earliest Doppler surveys revealed that the occurrence of giant planets with periods shorter than a few years is a steeply rising function of the host star s metallicity (Santos et al. 2003; Fischer and Valenti 2005). This is widely interpreted as support for the theory of giant-planet formation via core accretion. The logic is that the rapid assembly of a massive solid core an essential step in the theory is easier to arrange in a metal-rich protoplanetary disk. Most recently, Petigura et al. (207b) used Kepler data to determine the best-fitting parameters of Γ P,z = d 2 N d lnp d lnz CPα z β, (5)

7 Planet Occurrence Rates 7 where z is the iron-to-hydrogen abundance relative to the solar value. For planets larger than Neptune they found β 4, a remarkably strong dependence. For smaller planets the metallicity dependence was found to be weaker, in agreement with previous studies (see, e.g., Mayor et al. 20; Buchhave et al. 202). Nevertheless, for periods shorter than about 0 days, even planets smaller than Neptune are associated with slightly elevated metallicity (Mulders et al. 206). This trend is not as easy to understand in terms of core accretion theory. Other properties The giant-planet population is distinguished by other features that are beyond the scope of this review, but are mentioned here for completeness. They have a broad range of orbital eccentricities (see, e.g., Udry and Santos 2007). Their occurrence seems to fall precipitously for masses above 0 M Jup, a phenomenon that is often called the brown dwarf desert (Grether and Lineweaver 2006; Sahlmann et al. 20; Triaud et al. 207). There is a significant population of two-planet systems in mean-motion resonances (Wright et al. 20). The rotation of the star can be grossly misaligned with the orbit of the planet, especially if the star is more massive than about.2 M (Triaud, this volume). These and other topics were reviewed recently by Winn and Fabrycky (205) and Santerne (this volume). occurrence, N / log P Santerne et al. (206) Mayor et al. (20) Cumming et al. (2008) orbital period [days] Fig. 3 Occurrence of giant planets ( > 0. M Jup ) as a function of orbital period, from two independent Doppler surveys (Cumming et al. 2008; Mayor et al. 20) and the Kepler transit survey (Santerne et al. 206).

8 8 Joshua N. Winn Smaller planets Overall occurrence About half of Sun-like stars have at least one planet with an orbital period shorter than a year, and asize in between those of Earth and Neptune. Planet formation theories generally did not predict this profusion of planets. Indeed some of the most detailed theories predicted that such close-orbiting super-earth or sub-neptune planets would be especially rare (Ida and Lin 2008). Their surprisingly high abundance has led to new theories in which small planets can form in very short-period orbits, rather than forming farther away from the star and then migrating inward (see, e.g., Hansen and Murray 202; Chiang and Laughlin 203). Doppler surveys provided our first glimpse at this population of planets, and then Kepler revealed them in vivid detail. For planets with periods shorter than 50 days and minimum masses 3-30 M, two independent Doppler surveys found the occurrence to be (5±5)% (Howard et al. 200) and (27±5)% (Mayor et al. 20). For this same period range, and planets with a radius between 2-4 R, the Kepler survey found the occurrence to be (3.0 ± 0.8)% (Howard et al. 202). The results of the two surveys are compatible, given reasonable guesses for the relation between planetary mass and radius (Howard et al. 202; Figueira et al. 202). Size, mass, and period The surveys also agree that within this range of periods and planet sizes, the occurrence rate is higher for the smallest planets (Howard et al. 200, 202): dn d lnm dn m 0.5, d lnr R 2. (6) For even smaller or longer-period planets, Kepler provides almost all the available information (Fressin et al. 203). Figure 4 shows some of the latest results. The period distribution dn/d lnp rises as P 2 between -0 days, before leveling off to a nearly constant value between days. Multiple-planet systems Small planets occur frequently in closely-spaced systems (Fabrycky et al. 204). The most fecund Kepler systems have 7 transiting planets, all with periods shorter than a year. The period ratios tend to be in the neighborhood of.5-3. In units of the mutual Hill radius, ( ) Min + M /3 ( ) out ain + a out a H, (7) 3M 2 which is more directly relevant to orbital stability, the typical spacing is 0-30 (Fang and Margot 203). A few percent of these systems exhibit mean-motion resonances, suggesting that the orbits have been sculpted by planet-disk gravitational interactions. These resonant systems also offer the gift of transit-timing variations (Agol & Fabrycky, this volume). These observable manifestations of planet-planet gravitational interactions sometimes enable measurements of planetary masses as well as orbital eccentricities and inclinations. Such studies and some other lines of evidence show that the compact multiple-planet systems tend to have orbits that are circular

9 Planet Occurrence Rates 6 9 FGK dwarfs (Petigura et al. 208) radius [R ] 8 ert 4 hot 2 ne ptu des Ne low detection probability 6 M dwarfs (Dressing et al. 205) radius [R ] low detection probability 0 orbital period [days] 00 Fig. 4 Planet occurrence around FGK dwarfs (top) and M dwarfs (bottom) based on Kepler data. The blue dots are a random sample of 03 planets drawn from the occurrence rate densities derived by (Petigura et al. 207b) and Dressing and Charbonneau (205). Compared to FGK stars, the M stars have a higher occurrence of small planets and a lower occurrence of giant planets, at least for periods shorter than 00 days. For the M dwarfs, occurrence rates for planets larger than 4 R were not reported because only four such planets were detected. (Hadden and Lithwick 204; Xie et al. 206; Van Eylen and Albrecht 205) and coplanar (Fabrycky et al. 204). Radius gap The radius distribution of planets with periods shorter than 00 days has an interesting feature: a dip in occurrence between.5-2 R (Fulton et al. 207; Van Eylen et al. 207). Such a feature had been seen in theoretical calculations of the atmospheric erosion of low-mass planets due to the intense irradiation from the host star (Owen and Wu 203; Lopez and Fortney 203). Thus the radius gap seems to be a precious example in exoplanetary science of a prediction fulfilled, with many implications for the structures and atmospheres of close-orbiting planets (Owen and Wu 207). Hot Neptunes As mentioned above, the occurrence dn/d ln P changes from a ris< 0 days to a more constant value for P = 0-00 days. The ing function for P critical period separating these regimes is longer for larger planets. The effect is to

10 0 Joshua N. Winn 0.2 Number of Planets per Star (Orbital period < 00 days) typical uncert Planet Size [Earth radii] Fig. 5 From Fulton et al. (207). Occurrence as a function of radius based on Kepler data, for planets with orbital periods shorter than 00 days. There is a valley in the occurrence rate density between about.5-2 R, possibly the result of the erosion of planetary atmospheres by high-energy radiation from the star. create a diagonal boundary in the space of lnr and lnp, above which the occurrence is very low (see Fig. 4). The same phenomenon is seen in Doppler data and has been called the hot Neptune desert (Mazeh et al. 206). The desert may be another consequence of atmospheric erosion, although other explanations have been offered. One clue is that those few hot Neptunes that do exist are strongly associated with metal-rich stars (Dong et al. 207; Petigura et al. 207b), making them similar to hot Jupiters and unlike small planets more generally (Buchhave et al. 202). The hot Neptunes are also similar to hot Jupiters in that they tend not to have planetary companions in closely-spaced coplanar orbits (Dong et al. 207). Earth-like planets The goal of exoplanetary science with the broadest appeal is measuring the occurrence of Earth-sized planets orbiting Sun-like stars within the habitable zone, the range of distances within which a rocky planet could plausibly have oceans of liquid water. The Kepler mission provided by far the best data that has ever been available for this purpose. However, even Kepler was barely sensitive to such planets. The number of detections is of order 0, depending critically on the definitions of Earth-sized, Sun-like and habitable zone. Thus the measurement of d 2 N Γ d lnp d lnr (8) P= yr, R=R

11 Planet Occurrence Rates may require extrapolation from the measurements for shorter periods and larger planets. The Kepler team has published a series of papers reporting steady advancements in efficiency of detection, elimination of false positives, and understanding of instrumental artifacts. Nevertheless the most recent effort to determine η found the data to be compatible with any value from 0.04 to.5, given the range of possible systematic errors (Burke et al. 205, see Figure 6). Since then the Kepler team and other groups have clarified the properties of the stars that were searched (Petigura et al. 207a; De Cat et al. 205), and the most recent installments by Twicken et al. (206) and Thompson et al. (207) quantified the sensitivity of the algorithms for planet detection and validation. These developments have brought us right to the threshold of occurrence rate calculations for Earth-like planets. Probability Density Γ Foreman-Mackey Optimistic Efficiency Pessimistic Efficiency Original KIC DV Rp Low Reliability High Reliability Petigura Dong Direct Baseline Trimmed Direct Baseline Extrapolated Baseline Youdin Log 0 [Γ ] Fig. 6 From Burke et al. (205). Estimates for Γ based on Kepler data. The orange histogram is the posterior probability distribution considering only the uncertainties from counting statistics and extrapolation. The other curves illustrate the effects of some systematic errors: uncertainty in the detection efficiency, orbital eccentricities, stellar parameters, and reliability of weak planet candidates. These systematic effects led to a range in Γ spanning an order of magnitude. Also shown are other estimates of Γ by Foreman-Mackey et al. (204); Petigura et al. (203); Dong and Zhu (203) and Youdin (20).

12 2 Joshua N. Winn Other types of stars Almost all the preceding results were based on studies of main-sequence stars with masses M, i.e., spectral types from K to late F. Stars with masses M, the M dwarfs, are not as thoroughly explored. However they are very attractive for planet surveys because their small masses and sizes lead to larger Doppler and transit signals, and because planets in the habitable zone have conveniently short orbital periods. M dwarfs have a relatively low occurrence of giant planets, at least for periods shorter than a few years. Cumming et al. (2008) showed that if planet occurrence is modeled by the functional form of Eqn. (3), then M dwarfs have 3-0 times fewer planets than FGK dwarfs, for masses exceeding 0.4 M Jup and periods shorter than 5.5 years. Likewise, Bonfils et al. (203) found the occurrence of planets with m = M Jup and P < 00 days to be , which is 2-3 times smaller than the equivalent number for FGK stars. On the other hand, M dwarfs exceed FGK dwarfs by a factor of 2-3 in the occurrence of smaller planets over the same range of periods (Howard et al. 202; Mulders et al. 205). This is based on Kepler data, which remains our best source of information on this topic despite the fact that only a few percent of the Kepler target stars were M dwarfs. Dressing and Charbonneau (205) performed the most comprehensive analysis to date, finding the mean number of planets of radius -4 R and P < 50 days to be 2.5 ± 0.2. Since the habitable zones occur at periods of weeks to months, the implication is that the nearest habitable-zone planets are surely around M dwarfs, probably within just a few parsecs. Some other comparisons with FGK dwarfs have been made. One similarity is that M dwarfs often have compact systems of multiple planets (Ballard and Johnson 206). Another is that the radius gap between.5-2 R and the hot Neptune desert appear to exist for M dwarfs (Hirano et al. 207), although these features occur for planets at lower levels of irradiation than those around higher-mass stars. Whether or not planet occurrence is associated with high metallicity for M dwarfs is unclear (see, e.g., Gaidos and Mann 204). Beyond the scope of this review, but nevertheless fascinating, are the occurrence rates that have been measured in surveys of other types of stars: evolved stars (Johnson et al. 200; Reffert et al. 205), stars in open clusters (Mann et al. 207) and globular clusters (Gilliland et al. 2000; Masuda and Winn 207), binary stars (Armstrong et al. 204), brown dwarfs (He et al. 207), white dwarfs (Fulton et al. 204), and neutron stars (Wolszczan 202; Kerr et al. 205). Future Prospects The volume of data produced by recent surveys will offer opportunities for progress for at least another few years. The ongoing struggle to measure the occurrence of Earth-like planets has already been mentioned. Another undeveloped area is the

13 Planet Occurrence Rates 3 determination of joint and conditional probabilities; for example, given a planet with radius R and period P, what is the chance of finding another planet around the same star with radius R 2 and period P 2? Conditional occurrence is probably more useful than overall occurrence for testing theories of planet formation. Only a few specific cases have been studied, such as the mutual radius distribution of neighboring planets (Ciardi et al. 203; Weiss et al. 207), and the tendency for hot Jupiters to have wider-orbiting companions (Huang et al. 206; Bryan et al. 206; Schlaufman and Winn 206). New data are also forthcoming. The Transiting Exoplanet Survey Satellite (TESS) is scheduled for launch in 208 (Ricker et al. 205). This mission was not designed for measuring planet occurrence. Rather, the main goal is to pluck low-hanging fruit: to discover short-period transiting planets around nearby and bright stars. Less well appreciated is that TESS should be superior to Kepler in measuring the occurrence of planets larger than Neptune with periods shorter than 0 days. When TESS was conceived in 2006, it was expected that the limitations of data storage and transmission would limit the search to 0 5 pre-selected stars, similar to the Kepler mission. By 203, it became clear that entire TESS images could be stored and transmitted with 30-minute time sampling. As a result, although 0 5 Sun-like stars will still be selected for special treatment (two-minute time sampling), it will be possible to search millions of stars for large, short-period planets. Indeed TESS will excel at finding very rare, large-amplitude, short-period photometric phenomena of all kinds. For smaller planets around Sun-like stars, it will be difficult to achieve an orderof-magnitude improvement over the existing data. There is more room for improvement in the study of low-mass stars, using new Doppler spectrographs operating at infrared wavelengths, and ground-based transit surveys focusing exclusively on lowmass stars. Particularly encouraging was the discovery of TRAPPIST-, a system of seven Earth-sized planets orbiting an ultra-cool dwarf that barely qualifies as a star (Gillon et al. 207). This system was found after searching 50 similar objects with a detection efficiency of around 60% (Burdanov et al., this volume; M. Gillon, private communication), and the transit probability for the innermost planet is 5%. This suggests the occurrence of such systems is approximately ( ) = 0.7. Thus, while TRAPPIST- seems extraordinary, it may represent a typical outcome of planet formation around ultra-cool dwarfs. In the decades to come, the domains of all the planet detection techniques including astrometry, gravitational microlensing, and direct imaging will begin overlapping. Some efforts have already been made to construct occurrence models based on data from several very different techniques (see, e.g. Clanton and Gaudi 206). We can look forward to a much more holistic view of the occurrence of planets around other stars, barring any civilization-destroying cataclysm. Cross-References Populations of extrasolar giant planets from transit and radial velocity surveys

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